In the manufacture of methanesulfonic acid, for example in processes of the type disclosed in U.K. Pat. No. 1,350,328, an aqueous product which normally contains between about 15-60% by weight and higher amounts of water is obtained. In order to prepare an anhydrous product (&lt;2 wt. % water), useful, for example, as a reaction medium in the preparation of aromatic peroxy acids where excess water retards the reaction, the water must be removed from methanesulfonic acid while minimizing the formation of decomposition product e.g., methyl methanesulfonate, a known carcinogen. In the past, a two step distillative purification process.sup.1 for lower alkanesulfonic acids was used. However, this practice led to products which were partially decomposed and contained undesirable amounts of water. More recently, a superior process has been developed.sup.2 utilizing a "falling film" evaporator column which, at reduced energy requirements, provides a substantially anhydrous lower alkanesulfonic acid devoid of detectable amounts of harmful decomposition products. While this recently developed process provides much improved results, still further improvements in both the dryness of the product and productivity of the process are desirable. FNT .sup.1. See U.S. Pat. No. 4,035,242 and U.K., Pat. No. 1,350,328 FNT .sup.2. See U.S. Pat. No. 4,450,047